Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Introduction to Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Logarithms

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1:01 minutes

Problem 85

Textbook Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log 33)

Verified step by step guidance

Recognize that the expression is in the form of \( a^{\log_a b} \).

Recall the property of logarithms: \( a^{\log_a b} = b \).

Identify \( a = 10 \) and \( b = 33 \) in the expression \( 10^{\log 33} \).

Apply the property: \( 10^{\log_{10} 33} = 33 \).

Conclude that the expression simplifies to 33.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential and Logarithmic Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent, while logarithmic functions are the inverse operations of exponentials. Understanding the relationship between these two types of functions is crucial for simplifying expressions like 10^(log 33), as it allows us to manipulate the expression using properties of logarithms.

Properties of Logarithms

Logarithms have specific properties that simplify calculations, such as the fact that a^log_a(b) = b for any positive a. This property is essential for evaluating expressions involving logarithms, as it allows us to convert between exponential and logarithmic forms, making it easier to find the value of expressions like 10^(log 33).

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power to obtain a given value. In the expression log 33, the base is implicitly 10 (common logarithm). Recognizing the base is important for understanding how to evaluate expressions involving logarithms and exponentials, particularly when simplifying expressions like 10^(log 33).